A342458 -id:A342458 - cp's OEIS frontend

A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

1, 3, 4, 1, 6, 1, 8, 3, 2, 1, 12, 1, 14, 3, 8, 1, 18, 1, 20, 3, 2, 1, 24, 1, 2, 3, 4, 1, 30, 1, 32, 3, 2, 1, 12, 1, 38, 3, 8, 1, 42, 1, 44, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 62, 3, 32, 1, 6, 1, 68, 3, 2, 1, 72, 1, 74, 3, 4, 1, 6, 1, 80, 9, 2, 1, 84, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 24, 1, 98, 3, 4, 1, 102

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001615, A342916, A342917.
Cf. also A049559, A342458.
After n=1 differs from A143771 for the first time at n=44, where a(44) = 9, while A143771(44) = 3.

psi[n_] := If[n==1, 1, Times @@ ((#1+1)*#1^(#2-1)& @@@ FactorInteger[n])];
a[n_] := GCD[n+1, psi[n]];
Array[a, 105] (* Jean-François Alcover, Dec 22 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A342915(n) = gcd(1+n,A001615(n));

a(n) = gcd(1+n, A001615(n)).

a(n) = (1+n) / A342916(n) = A001615(n) / A342917(n).

A348492 Greatest common divisor of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 1, 1, 4, 1, 5, 1, 4, 3, 1, 1, 8, 1, 3, 1, 16, 1, 21, 1, 24, 5, 1, 1, 4, 5, 15, 27, 8, 1, 1, 1, 16, 7, 1, 3, 12, 1, 3, 1, 4, 1, 1, 1, 24, 3, 5, 1, 16, 7, 15, 5, 8, 1, 81, 1, 4, 1, 1, 1, 4, 1, 3, 3, 64, 9, 1, 1, 24, 1, 1, 1, 12, 1, 3, 5, 8, 3, 1, 1, 16, 27, 1, 1, 4, 11, 15, 1, 140, 1, 3, 5, 24, 1, 1, 3, 16, 1, 7

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A003557, A018804, A348493, A348494.

Cf. also A342413, A342458, A348495.

Array[GCD[Total@ GCD[#, Range[#]], # Total[#2/#1 & @@@ FactorInteger[#]]] &, 98] (* Michael De Vlieger, Oct 21 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348492(n) = gcd(A003415(n), A018804(n));

a(n) = gcd(A003415(n), A018804(n)).
For n > 1, a(n) = A003415(n) / A348493(n).
a(n) = A003557(n) * A348494(n).

A342919 a(n) = A003415(n) / gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 1, 1, 7, 1, 2, 1, 3, 1, 4, 1, 7, 1, 2, 5, 13, 1, 11, 1, 5, 3, 2, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 2, 13, 25, 1, 7, 1, 1, 5, 2, 1, 3, 2, 23, 11, 31, 1, 23, 1, 11, 17, 2, 3, 61, 1, 2, 13, 59, 1, 13, 1, 13, 11, 2, 3, 71, 1, 11, 1, 43, 1, 31, 11, 15, 4, 35, 1, 41, 5, 2, 17, 49, 1, 17, 1, 11, 25

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001615, A003415, A342001, A342458, A342459.

Cf. also A342414.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342919(n) = { my(u=A003415(n)); (u/gcd(u, A001615(n))); };

a(n) = A003415(n) / A342458(n) = A003415(n) / gcd(A001615(n), A003415(n)).

a(n) = A342001(n) / A342459(n).

A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

Original entry on oeis.org

8, 81, 108, 2500, 2700, 3375, 5292, 13068, 15625, 18252, 31212, 38988, 57132, 67228, 90828, 94500, 103788, 147852, 181548, 199692, 231525, 238572, 303372, 375948, 401868, 484812, 544428, 575532, 674028, 713097, 744012, 855468, 1016172, 1058841, 1101708, 1145772

Offset: 1

Paolo P. Lava, Mar 29 2018

If n = Product_{k=1..j} p_k ^ i_k with each p_k prime, then psi(n) = n * Product_{k=1..j} (p_k + 1)/p_k and n' = n*Sum_{k=1..j} i_k/p_k.
Thus every number of the form p^(p+1), where p is prime, is in the sequence.
The sequence also contains every number of the form 108*p^2 where p is a prime > 3, or 108*p^3*(p+2) where p > 3 is in A001359. - Robert Israel, Mar 29 2018

			5292 = 2^2 * 3^3 * 7^2.
n' = 5292*(2/2 + 3/3 + 2/7) = 12096,
psi(n) = 5292*(1 + 1/2)*(1 + 1/3)*(1 + 1/7) = 12096.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A001359, A001615, A003415, A166374, A342458. A345005 (gives the odd terms).

Subsequence of A345003.

with(numtheory): P:=proc(n) local a,p; a:=ifactors(n)[2];
if add(op(2,p)/op(1,p),p=a)=mul(1+1/op(1,p),p=a) then n; fi; end:
seq(P(i),i=1..10^6);

selQ[n_] := Module[{f = FactorInteger[n], p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, f}] == Sum[{p, e} = pe; (n/p)e, {pe, f}]];
Select[Range[10^6], selQ] (* Jean-François Alcover, Oct 16 2020 *)

dpsi(f) = prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
ader(n, f) = sum(i=1, #f~, n/f[i, 1]*f[i, 2]);
isok(n) = my(f=factor(n)); dpsi(f) == ader(n, f); \\ Michel Marcus, Mar 29 2018

Solutions of the equation n' = psi(n).

A342459 a(n) = gcd(A048250(n), A342001(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 1, 3, 8, 1, 1, 1, 1, 6, 2, 1, 1, 1, 2, 3, 1, 8, 1, 1, 1, 1, 2, 1, 12, 2, 1, 3, 8, 1, 1, 1, 1, 12, 1, 1, 1, 2, 2, 9, 4, 14, 1, 3, 8, 1, 2, 1, 1, 2, 1, 3, 1, 3, 6, 1, 1, 18, 2, 1, 1, 1, 1, 3, 1, 20, 6, 1, 1, 2, 4, 1, 1, 2, 2, 3, 8, 1, 1, 1, 4, 24, 2, 1, 24, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001615, A003415, A003557, A048250, A342001, A342458, A342919.

Cf. also A066086, A342416.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A342459(n) = gcd(A048250(n), A342001(n));

a(n) = A342458(n) / A003557(n) = gcd(A048250(n), A342001(n)).

a(n) = A342001(n) / A342919(n).

A348028 Greatest common divisor of A003415 (arithmetic derivative) and sigma, the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 8, 1, 1, 3, 1, 6, 2, 1, 1, 4, 1, 3, 1, 8, 1, 1, 1, 1, 2, 1, 12, 1, 1, 3, 8, 2, 1, 1, 1, 12, 39, 1, 1, 4, 1, 3, 4, 14, 1, 3, 8, 4, 2, 1, 1, 4, 1, 3, 1, 1, 6, 1, 1, 18, 2, 1, 1, 39, 1, 3, 1, 20, 6, 1, 1, 2, 1, 1, 1, 4, 2, 3, 8, 20, 1, 3, 4, 24, 2, 1, 24, 4, 1, 1, 3, 7, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Sep 25 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415.

Cf. also A342413, A342458, A343226.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A348028(n) = gcd(sigma(n), A003415(n));

a(n) = gcd(A000203(n), A003415(n)).

cp's OEIS Frontend

A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A348492 Greatest common divisor of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

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A342919 a(n) = A003415(n) / gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

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A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

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A342459 a(n) = gcd(A048250(n), A342001(n)).

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A348028 Greatest common divisor of A003415 (arithmetic derivative) and sigma, the sum of divisors function.

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